Nonequilibrium first-order phase transition in coupled oscillator systems with inertia and noise

Abstract

We study the dynamics of a system of coupled oscillators of distributed natural frequencies, by including the features of both thermal noise, parametrized by a temperature, and inertial terms, parametrized by a moment of inertia. For a general unimodal frequency distribution, we report here the complete phase diagram of the model in the space of dimensionless moment of inertia, temperature, and width of the frequency distribution. We demonstrate that the system undergoes a nonequilibrium first-order phase transition from a synchronized phase at low parameter values to an incoherent phase at high values. We provide strong numerical evidence for the existence of both the synchronized and the incoherent phase, treating the latter analytically to obtain the corresponding linear stability threshold that bounds the first-order transition point from below. In the limit of zero noise and inertia, when the dynamics reduces to the one of the Kuramoto model, we recover the associated known continuous transition. At finite noise and inertia but in the absence of natural frequencies, the dynamics becomes that of a well-studied model of long-range interactions, the Hamiltonian mean-field model. Close to the first-order phase transition, we show that the escape time out of metastable states scales exponentially with the number of oscillators, which we explain to be stemming from the long-range nature of the interaction between the oscillators.

Spontaneous synchronisation and nonequilibrium statistical mechanics of coupled phase oscillators

“…[29,30,31,32,33,20]. Inclusion of inertia elevates the first-order Kuramoto dynamics to one that is second order in time, while noise accounts for temporal fluctuations of the natural frequencies.…”

“…g(ω) = δ(ω) the dynamics (44) is that of the BMF model with an equilibrium stationary state. For other g(ω), it may be shown that the dynamics (44) violates detailed balance, leading to a NESS [33].…”

“…Equation (33) being a representative Langevin dynamics may be studied by employing the corresponding tool of analysis usual in statistical physical studies, namely, the Fokker-Planck equation [27,28] for the time evolution of the single-oscillator probability density ρ(θ, ω, t) defined above. This equation may be derived straightforwardly for the dynamics (33), and has the form…”

“…Let us remark that all phase transitions for σ = 0 are in NESSs. The first-order nature of the phase transition becomes evident on analyzing results of N -body simulations of the dynamics (44) for a representative g(ω), for example, a Gaussian distribution g(ω) = exp(−ω 2 /2)/ √ 2π [33,20]. For given values of m and T , an initial state, which has all the oscillators at θ = 0 and angular velocities v i 's sampled from a Gaussian distribution with zero mean and standard deviation ∝ T , was first allowed to equilibrate at σ = 0.…”

“…This result in proved in Section 4.5. (44), in which the thick red secondorder critical lines denote the continuous transitions mentioned above [33,20]. For m, σ, T all 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 (44) in the three-dimensional space of the parameters, the dimensionless moment of inertia m, the temperature T , and the width of the frequency distribution σ.…”